If speed of water in river is 4 m/s and speed of swimmer with respect to water is 3 m/s, then in which direction the swimmer must swim so that he will reach directly opposite end?

Options

A. `127^(@)` with direction of river flow
B. `60^(@)` with direction of river flow
C. `143^(@)` with direction of river flow
D. Swimmer will never reach directly opposite end

Answer

To solve the problem of determining the direction a swimmer must swim in order to reach the directly opposite bank of a river, we can follow these steps:

### Step 1: Understand the Given Information
- Speed of water in the river (V_r) = 4 m/s
- Speed of swimmer with respect to water (V_s) = 3 m/s

### Step 2: Analyze the Situation
The swimmer is trying to swim directly across the river. However, the current of the river will push the swimmer downstream. To reach the opposite bank directly, the swimmer must swim at an angle upstream against the current.

### Step 3: Set Up the Velocity Components
Let’s denote:
- The angle at which the swimmer swims upstream as θ.
- The component of the swimmer's velocity across the river (perpendicular to the current) is V_s * cos(θ).
- The component of the swimmer's velocity against the current (parallel to the current) is V_s * sin(θ).

### Step 4: Establish the Condition for Reaching Directly Opposite
For the swimmer to reach the directly opposite point, the upstream component of the swimmer's velocity must equal the speed of the current:
\[ V_s \cdot \sin(θ) = V_r \]

### Step 5: Substitute the Known Values
Substituting the known values into the equation:
\[ 3 \cdot \sin(θ) = 4 \]

### Step 6: Solve for sin(θ)
Rearranging the equation gives:
\[ \sin(θ) = \frac{4}{3} \]

### Step 7: Analyze the Result
Since the value of sin(θ) cannot exceed 1, this indicates that there is no angle θ that satisfies this condition. Therefore, the swimmer cannot swim at a speed that allows him to reach the directly opposite bank.

### Conclusion
The swimmer will never reach the directly opposite bank of the river because his speed with respect to the water is less than the speed of the water current.

### Final Answer
The swimmer will never reach directly opposite.

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